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Assume $H \subset G$ is a closed subgroup, and the inclusion of groups is a homotopy equivalence. If $X$ is a CW complex and $E$ is a principal $G$-bundle over $X$, is there a principal $H$-bundle $E'$ over $X$ where $E'(G) := (E' \times G)/H$ is isomorphic to $E$, where $H$ acts on $G$ by left multiplication?

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The answer is yes. This is the same as asking if the classifying map $X \to BG$ factors up to homotopy as $X \to BH \to BG$. In your situation, since $H \to G$ is a homotopy equivalence, $BH \to BG$ will be a homotopy equivalence. Getting an actual homotopy equivalence and not just a weak equivalence depends on your model for $BG$, This question and this question have more information.

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